The Dvoretzky-Wald-Wolfowitz Theorem and Purification in Atomless Finite-action Games

Transcription

1 he Dvoretzky-Wald-Wolfowitz heorem and Purification in tomless Finite-action Games M. li Khan, Kali P. Rath and Yeneng Sun ugust 2005 bstract In 1951, Dvoretzky, Wald and Wolfowitz (henceforth DWW) showed that corresponding to any mixed strategy into a finite action space, there exists a pure-strategy with an identical integral with respect to a finite set of atomless measures. DWW used their theorem for purification: the elimination of randomness in statistical decision procedures and in zero-sum two-person games. In this short essay, we apply a consequence of their theorem to a finite-action setting of finite games with incomplete and private information, as well as to that of large games. In addition to simplified proofs and conceptual clarifications, the unification of results offered here re-emphasizes the close connection between statistical decision theory and the theory of games. Journal of Economic Literature Classification Numbers: C07, DO5. Key Words: DWW heorem, atomless, mixed and pure strategies equilibria, randomization, purification. Running itle: he DWW heorem and tomless Games first draft of this paper was completed when Khan and Rath were visiting the Institute for Mathematical Sciences at the National University of Singapore in ugust 2003; they thank the Institute for supporting their visit. preliminary version was presented at the Midwest Economic heory Conference held at Indiana University, Bloomington in October 2003; the authors are grateful to Eric Balder, Robert Becker, William homson and Myrna Wooders for questions and helpful suggestions. Department of Economics, he Johns Hopkins University, Baltimore, MD 21218, US. Department of Economics and Econometrics, University of Notre Dame, Notre Dame, IN 46556, US. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore ; and Department of Economics, National University of Singapore, 1 rts Link, Singapore

2 1 Introduction Dvoretzky, Wald and Wolfowitz (henceforth DWW), in a theorem proved in DWW (1951b, heorem 4), and announced in DWW (1950, heorem 1) and in DWW (1951a, heorem 2.1), showed that corresponding to any mixed strategy into a finite action space, 1 there exists a pure-strategy such that the two strategies have the same integral with respect to a finite set of atomless measures. hey proved their theorem using the Lyapunov theorem for vector measures, and formulated, as a simple consequence of the theorem, a general purification principle that any mixed strategy can be purified to yield the same expected payoffs and distributions. he general principle is then applied to the purification of statistical decision procedures, and of mixed strategies in two-person zero-sum games with finite action sets. 2 In this paper, we observe that the general purification principle of DWW is applicable to finite games with incomplete information in Radner-Rosenthal (1982) as well as to large non-anonymous games in Schmeidler (1973), both presented in a finite-action setting. he relevance of the DWW insight to the purification problem in finite games with incomplete and diffuse information was already suggested in Radner-Rosenthal (1982, Footnote 3] 3 and in Milgrom-Weber (1985, Section 5). 4 However, the purification result in Milgrom-Weber (1985) does not follow directly from the original result in DWW (1951a) as claimed therein, but from a new corollary of the DWW heorem formulated here. In this context, we also clarify different notions of purification implicit in DWW (1951a), Milgrom-Weber (1985) and Radner-Rosenthal (1982), and prove some results for games with incomplete information, based on what we term strong purification. his concept is stronger than all of the purification concepts in the relevant literature. With the alternative mathematical framework in place, it is also straightforward to derive the symmetrization result on equilibria in large anonymous games, as in Khan-Sun (1991) and Mas-Colell (1984). We point out that the derivation of each of the purification results in this paper is not available in the literature, and their directness and simplicity may perhaps be surprising. o summarize, this essay, in giving a central location to the DWW theorem, re-emphasizes the intimate connection between statistical decision theory and the theory of games, an interdisciplinary thrust clearly evident in the classical papers, and one that may possibly suggest fruitful new questions for both subjects. 5 2 he DWW heorem Let IR n denote n-dimensional Euclidean space and IR n + its non-negative orthant. For any two measurable spaces (X, X ) and (Y, Y), Meas(X, Y ) denotes the space of (X, Y)-measurable functions. Where needed, IR n will be endowed with its Borel σ-algebra. We reproduce the DWW theorem for the reader s convenience. 6 1 We alert the reader to the fact that what we term a mixed strategy here is called a behavioral strategy in Radner- Rosenthal (1982) and Milgrom-Weber (1985), while a mixed strategy there carries a different meaning. 2 See respectively DWW (1950, heorems 5 and 6), DWW (1951a, heorems 3.1 and 3.2, Section 4, heorems 5.1 and 5.2) on statistical decision procedures, and DWW (1950, heorems 2 and 3), DWW (1951a, Section 9) on two-person zero-sum games. In DWW (1950, heorem 4) and DWW (1951a, Section 8), they also consider approximate purification, an issue that is outside the scope of this paper. 3 Radner-Rosenthal (1982, Footnote 3) noted that their method for the proof of their heorem 1 is reminiscent of the DWW heorem without giving a proof based on it. 4 However, the relevance of the DWW insight to the issue of purification in large games has apparently not been noticed in a large and growing literature; see the references in Khan-Sun (2002). 5 In this connection, we refrain from the consideration of sequential statistical procedures, and, as noted above, from issues relating to approximate purification, see DWW (1950, 1951a) and Wald-Wolfowitz (1951). 6 See DWW (1951a, heorem 2.1) and the proof of heorem 4 in DWW (1951b). 1

3 heorem [DWW] Let (, ) be a measurable space; µ k, k = 1,, m, finite, atomless measures on (, ); and f i Meas(, IR + ), i = 1,, n, such that for all t, n i=1 f i(t) = 1. hen there exist measurable functions fi Meas(, {0, 1}), i = 1,, n, such that n i=1 f i (t) = 1 for all t and f i (t)dµ k (t) = fi (t)dµ k (t) for all k = 1,, m, and i = 1,, n. For a finite set, let M() be the set of probability measures on, and since it can be embedded in a finite-dimensional Euclidean space, we shall assume it to be endowed with its Borel σ-algebra. We shall now present a corollary of the DWW heorem that is used to derive a variety of purification results for atomless games. It also has independent interest. Note that ssertions (1) and (2) in the corollary are stated separately in DWW (1951a, Section 3) as heorems 3.1 and 3.2 for atomless distribution functions; we put them together for general atomless probability measures. ssertions (2) and (3) are quoted in Milgrom-Weber (1985, last paragraph on page 629) as a result in DWW (1951a). However, ssertion (3) is not proved in DWW (1951a) as is claimed in Milgrom-Weber (1985); we prove it here and put it together with ssertions (1) and (2). he proof of the corollary we present here adapts the proof in DWW (1951a, Section 3) along with the additional trick of applying the DWW heorem to a specially chosen function v l+n+1 (a, t) = 1 \F (t) for proving ssertion (3). Corollary 1 Let (, ) be a measurable space; µ k, k = 1,, m, atomless probability measures on (, ); a finite set represented as {a 1,, a n }; v j, j = 1,, l, elements of Meas(, IR) that are integrable with respect to each of the measures µ k ; and g Meas(, M()). Let g(t; S) represent the value of the probability measure g(t) at S and g(t; da) the integration operator with respect to it. hen there exists g Meas(, ) such that for all k = 1,, m, 1. for all j = 1,, l, v j(a, t)g(t; da)dµ k (t) = v j(g (t), t)dµ k (t); 2. for all B, g(t; B)dµ k(t) = µ k g 1 (B); 3. g (t) {a i : g(t; {a i }) > 0} supp g(t) for µ k -almost all t. Proof: We shall supplement the l given functions v j by n + 1 additional functions. owards this end, let v j be the indicator function 1 {aj l } on for j = l + 1,, l + n and v l+n+1 (a, t) = 1 \F (t), where F (t) = {a : g(t; {a}) > 0}, \ F (t) denotes set-theoretic subtraction, and for a set B, 1 B is the indicator function of the set B. Certainly v j is bounded and in Meas(, IR) for each j = l + 1,, l + n + 1. For notational convenience, let l + n + 1 = q. We first establish that there exists g Meas(, ) such that v j (a, t)g(t; da)dµ k (t) = v j (g (t), t)dµ k (t) for all k = 1,, m and j = 1,, q. (1) owards this end, define for any W, ν ijk (W ) = W v j(a i, t)dµ k (t) for all i = 1,, n, j = 1,, q, and k = 1,, m. Certainly ν ijk are atomless finite (signed) measures. Define for each i = 1,, n, f i (t) = g(t)({a i }). Certainly n i=1 f i(t) = 1. On applying the DWW heorem to ν ijk and to f i, we are guaranteed the existence of functions fi Meas(, {0, 1}), i = 1,, n such that n i=1 f i (t) = 1 for all t, and f i (t)dν ijk (t) = fi (t)dν ijk (t) for all i = 1,, n; j = 1,, q and k = 1,, m. (2) 2

4 On substituting the values of ν ijk in (2), we obtain v j (a i, t)f i (t)dµ k (t) = v j (a i, t)fi (t)dµ k (t) for all i = 1,, n; j = 1,, q and k = 1,, m, which, in turn, implies that for any j = 1,, q, and k = 1,, m, v j (a i, t)f i (t)dµ k (t) = i=1 v j (a i, t)fi (t)dµ k (t). (3) i=1 For each i = 1,, n, let i = {t : fi (t) = 1} and g :, g (t) = a i for all t i. It is clear from the properties of fi that { i } is a measurable decomposition of, and that therefore g is well-defined. hen, Equation (3) implies that for any j = 1,, q and k = 1,, m, v j (a, t)g(t; da)dµ k (t) = = = v j (a i, t)g(t; {a i }))dµ k (t) = i=1 v j (a i, t)fi (t)dµ k (t) = i=1 v j (g (t), t)dµ k (t) = i i=1 i=1 v j (a i, t)f i (t)dµ k (t) i=1 i v j (a i, t)dµ k (t) v j (g (t), t)dµ k (t), and the proof of the claim is complete. Now, on choosing j = 1,, l, in Equation (1), we obtain the first assertion of the corollary. Next, to show that the distribution induced by g (t) is the same as that induced by g on for any of the m measures, we apply Equation (1) to the function v j, j = l + 1,, l + n, to obtain that for any k = 1,, m, g(t; {a i })dµ k (t) = 1 {ai}(a)g(t; da)dµ k (t) = 1 {ai}(g (t))dµ k (t) = µ k g 1 ({a i }) holds for all i = 1,, n. his implies ssertion (2). Finally, on choosing j = l + n + 1, in Equation (1), we obtain for any k = 1,, m, 1 \F (t) (a)g(t; da)dµ k (t) = 1 \F (t) (g (t))dµ k (t). Since the first element of the equality is zero by construction, so is the second element, and this implies that 1 \F (t) (g (t)) equals zero for µ k -almost all t. hus, for µ k -almost all t, g (t) supp g(t). 3 Finite Games with Incomplete Information: Conditional Independence In this section, we consider finite games with incomplete information as formulated by Milgrom-Weber (1985). game with incomplete information Γ MW consists of a finite set of l players and an information space available to them. Each player i is endowed with a finite action set i (denote the product space Π l j=1 j by ), a measurable space ( i, i ) representing his possible types, and a payoff function u i : 3

5 0 i IR. Let the measurable space ( 0, 0 ), 0 = {t 01,, t 0m }, represent the space of common states that affect the payoffs of all the players. he product measurable space (, ) (Π l j=0 j, Π l j=0 j) equipped with a probability measure η constitutes the information space of the game. ssume that for any a, u i (a, t 0, t i ) is integrable on (,, η). 7 For each t 0k 0, k = 1,, m, let η( ; t 0k ) denote the conditional probability 8 on the space (Π l j=1 j, Π l j=1 j). Following Milgrom-Weber (1985), we shall assume that for each i = 1,, l, the marginal η i ( ; t 0k ) of η( ; t 0k ) on the space ( i, i ) is atomless and that η( ; t 0k ) = Π l i=1 η i( ; t 0k ). he latter condition is simply a formalization of the intuitive statement that conditional on 0, the players types are independent. It is also abbreviated to as conditional independence of probability measures. We shall denote the measure η i ( ; t 0k ) by µ ik. he marginal of η on ( 0, 0 ) is denoted by η 0, and η 0 (t 0k ) by µ 0k for each k = 1,, m. For any player i, a mixed strategy is an element of Meas( i, M( i )), and a pure strategy is an element of Meas( i, i ). pure strategy f i Meas( i, i ) can also be viewed as a mixed strategy g i by taking g i (t i ) to be the Dirac measure δ fi(t i) at f i (t i ) for each t i i. mixed (pure) strategy profile is a collection g = {g i } l i=1 of mixed (pure) strategies that specify a mixed (pure) strategy for each player. For a player i = 1,, l, we shall use the following (conventional) notation: i = Π 1 j l,j i j, i = Π 1 j l,j i j, a = (a i, a i ) for a, (t 1,..., t l ) = (t i, t i ) for (t 1,..., t l ) Π l j=1 j, and g = (g i, g i ) for a strategy profile g. 9 ssume that the players play the mixed strategy profile g = {g i } l i=1. hen, the resulting expected payoff for player i can be written as U i (g) = U i (g 1,, g l ) = u i (a, t i, t 0 )g 1 (t 1 ; da 1 ) g l (t l ; da l )dη = m µ 0k k=1 i i v g ik (a i, t i )g i (t i ; da i )dµ ik (t i ), (4) where v g ik (a i, t i ) (which depends on the mixed strategy profile g) equals t i i a i i u i (a i, a i, t i, t 0k )Π j i g j (t j ; da j )dπ j i µ jk (t j ), (5) and the second equality in Equation (4) is obtained from the assumption of conditional independence. he fact that the functions v g ik (a i, t i ) are in Meas( i i, IR) is a consequence of Fubini s theorem. 10 For each j = 1,, l, denote the measure j g j (t j, )dµ jk (t j ) on j by γ gj jk. hen we can rewrite Equation (5) as v g ik (a i, t i ) = u i (a i, a i, t i, t 0k )dπ j i γ gj jk (a i). (6) a i i hus, the i-th player s expected payoff depends on the actions of the other players only through the conditional distribution of their strategies induced on their action spaces. In the following, we define different types of equivalent strategy profiles, and this furnishes different types of purification concepts. 7 boundedness condition on the payoffs is assumed in Milgrom-Weber (1985, p. 623). 8 It always exists since 0 is finite. 9 From now on, without any ambiguity, we shall abbreviate Π 1 j l,j i to Π j i. 10 For the details of this theorem, see, for example, sh (1972). 4

6 Definition 1 Let g = {g i } l i=1 and g = {gi }l i=1 be two mixed strategy profiles. 1. he strategy profile g is said to be payoff equivalent to g if U i (g) = U i (g ) for all players i = 1,..., l. 2. he strategy profiles g and g are said to be strongly payoff equivalent if (i) they are payoff equivalent; (ii) for any player i and any given mixed strategy g i Meas( i, M( i )), U i (g i, g i) = U i (g i, g i ). 3. For a player i, the strategy g i is said to be distribution equivalent to the strategy gi if they have the same conditional distribution on the action space i in the sense that for all k = 1,..., m, i g i (t i ; )dµ ik (t i ) = i gi (t i; )dµ ik (t i ). he strategy profile g is said to be distribution equivalent to the strategy profile g if g i is distribution equivalent to gi for all players i = 1,..., l. 4. For a player i, the pure strategy gi is said to be strongly distribution equivalent to the strategy g i if gi is distribution equivalent to g i, and for each k = 1,, m, gi (t i) supp g i (t i ) for µ ik -almost all t i i. he pure strategy profile g is said to be strongly distribution equivalent to the strategy profile g if gi is strongly distribution equivalent to g i for all players i = 1,..., l. 5. pure strategy profile g is said to be a strong purification of the strategy profile g if g is both strongly payoff equivalent and strongly distribution equivalent to g. Item (2)(ii) above says that the expected payoff of player i from an arbitrary mixed strategy is the same irrespective of whether his opponents play g i or g i. It is thus clear that if two strategy profiles are strongly payoff equivalent and one is an equilibrium of the game Γ MW, then the other is also an equilibrium. he following two examples establish, in the simple context of a single player games, that payoff equivalence and distribution equivalence are independent concepts. he first example shows that strong distribution equivalence does not imply payoff equivalence. Example 1 Let (,, λ) be the unit Lebesgue interval. Let the action set = {0, 1}. Let ν be the uniform distribution on, which is to say ν({0}) = 1/2 = ν({1}), and let g Meas(, M()) such that g(t) = ν for all t. Let g : be such that { g 1 for 0 t < 1/2 (t) = 0 for 1/2 t 1 It is clear that g (t) supp g(t) for all t, and the induced distribution λg 1 on is identical to ν. Hence, g is strongly distribution equivalent to the pure strategy g. Next, let u : IR be such that for all t, { 0 for a = 1, u(a, t) = g (t) for a = 0. We can now compute the two expressions: U(g) = u(a, t)g(t; da)dλ(t) = (1/2) U(g ) = u(a, t)δ g (t)(da)dλ(t) = [u(0, t) + u(1, t)]dλ(t) = (1/2) u(g (t), t)dλ(t) = 5 1/2 0 u(1, t)dλ(t) + g (t)dλ(t) = 1/4, 1 1/2 u(0, t)dλ(t) = 0.

7 hus, g is not payoff equivalent (thus not strongly payoff equivalent) to g. he second example shows that (strong) payoff equivalence does not imply distribution equivalence; and furthermore, (strong) payoff equivalence and distribution equivalence do not imply strong distribution equivalence. Example 2 Let (,, λ) be the unit Lebesgue interval. Let the action set = {0, 1}. Let u : IR be the constant payoff function with value 1, and ν the uniform distribution on (ν({0}) = 1/2 = ν({1})), and let g Meas(, M()) be such that { ν for 0 t < 1/2 g(t) = δ {1} for 1/2 t 1 Define g, g : such that g (t) = 0 for all t, and { g 1 for 0 t < 3/4 (t) = 0 for 3/4 t 1 hen, g is payoff equivalent to g but not distribution equivalent to g; g is both payoff and distribution equivalent to g but not strongly distribution equivalent to g. Various notions of purification have been used in the literature. Both payoff equivalence and distribution equivalence are used for the purification of statistical decision procedures in DWW (1951a, p. 2). 11 Payoff equivalence is used for purification of mixed strategies in two-person zero-sum games in DWW (1950, heorems 2 and 3, p. 257) and DWW (1951a, p. 20), and of mixed strategy equilibria in finite games with incomplete information in Radner-Rosenthal (1982, p. 403). Milgrom-Weber (1985) defined a notion of purification in Section 5 that is weaker than the strong distribution equivalence in Definition 1; they also claimed in the last paragraph on page 629 and in the proof of their heorem 4 that DWW proved in DWW (1951a) that any (possibly non-equilibrium) mixed strategy profile has a strongly distribution equivalent pure strategy profile. We can make two observations in this connection. he first point is that this latter result is not contained in DWW (1951a) as claimed in Milgrom-Weber (1985) (though it follows from our Corollary 1 here). 12 he second point concerns payoff equivalence. Since expected payoffs provide a starting point for discussing players behavior in a game with incomplete information, a reasonable purification concept for a general mixed strategy profile ought at least to lead to equivalent expected payoffs; however, Example 1 shows that this (minimal) requirement does not follow from strong distribution equivalence. he following theorem provides a result in terms of strong purification. heorem 1 In the game Γ MW, every mixed strategy profile has a strong purification. Proof: Let g = (g 1,, g l ) be a mixed strategy profile for the game Γ MW. Fix any player i = 1,, l. For each k = 1,, m, compute µ ik and v g ik. Now apply Corollary 1 to the collection {( i, i ), {µ ik } m k=1, i, {v g ik }m k=1, g i } to obtain a pure strategy g i Meas( i, i ) such that for all k = 1,, m, 11 he notions of payoff equivalence and distribution equivalence here are called respectively equivalence and strong equivalence in the bottom of page 2 in DWW (1951a). 12 It is claimed in DWW (1951a, p. 6) that Equations (4.5) to (4.8) can be obtained by applying Equations (3.3) and (3.5) to the loss functions W (F i, d m1 m l, x). While (4.5), (4.6) and (4.8) can be obtained in this way, (4.7) cannot. he authors thank Zhixiang Zhang for this observation. 6

8 (i) i i v g ik (a i, t i )g i (t i ; da i )dµ ik (t i ) = i v g ik (g i (t i), t i )dµ ik (t i ); (ii) for all B i, i g i (t i ; B)dµ ik (t i ) = µ ik g 1 i (B) = γ gi ik ; (iii) g i (t i) supp g i (t i ) for µ ik -almost all t i i. Let g = (g1,, gl ). hen (ii) and (iii) above imply that the pure strategy profile g is strongly distribution equivalent to the strategy profile g. ll that remains to be shown is the strong payoff equivalence of g and g. owards this end, consider any mixed strategy g i Meas( i, M( i )). Denote (g i, g i) by g and (g i, g i ) by g. By Equations (4) and (6), the expected payoffs of player i with g and g are respectively given by m U i (g ) = µ 0k ik (a i, t i )g i(t i ; da i )dµ ik (t i ), (7) U i (g ) = k=1 m µ 0k k=1 i i i v g v g i ik (a i, t i )g i(t i ; da i )dµ ik (t i ), (8) where v g ik (a i, t i ) = a i i u i (a i, a i, t i, t 0k )dπ j i γ g j jk (a i). By (ii) above and the fact that g i = g i and g i = g j i, we have γg jk = γg j jk = γgj jk = γg j jk for all j = 1,..., l with j i and k = 1,..., m. Hence v g ik (, ) = vg ik (, ) for all k = 1,..., m. By Equations (7) and (8), we have U i(g ) = U i (g ), and also U i (g ) = m µ 0k v g ik (g i (t i ), t i )dµ ik (t i ). (9) i k=1 Hence, U i (g ) = U i (g) by Equation (4) and (i) above. herefore, g i is strongly payoff equivalent to g i. Since player i was chosen arbitrarily, the proof is finished. 4 Finite Games with Private Information: Mutual Independence Next, we turn to finite games with private information as formulated by Radner-Rosenthal (1982). We shall reformulate it as a special case of the game Γ MW considered in the last section. his allows a synthetic treatment of finite games with private information that is independent or conditionally independent. game with private information Γ RR consists of a finite set of l players, each of whom is endowed with a finite action set i (as above, denotes the product space Π l j=1 j), an information space constituted by a pair of l measurable spaces ( i, i ) and (S i, S i ) together with a probability measure µ on the product measurable space (S, S) (Π l j=1 ( j S j ), Π l j=1 ( j S j ), and finally, a payoff function u i : S i IR. For any point s = (t 1, s 1,, t l, s l ) S, and for any i = 1,, l, let (ζ i, σ i ) be the coordinate projections, which is to say that ζ i (s) = t i and σ i (s) = s i. We shall assume that for every player i, the distribution η i = µζ 1 i of ζ i is atomless, and that the random variables {ζ j : j i} together with the random variable ξ i (ζ i, σ i ) form a mutually independent set of random variables. ssume that for any a, u i (a, σ i ( )) is integrable on (S, S, µ). 7

9 Mixed (pure) strategies as well as mixed (pure) strategy profiles for the game Γ RR can be defined as in Section 3. For a given mixed strategy profile g = {g i } l i=1, the resulting expected payoff for player i is given by U i (g) = U i (g 1,, g l ) = S l u i (a, σ i (s))g 1 (ζ 1 (s); da 1 ) g l (ζ l (s); da l )dµ(s). 1 (10) Denote the measure S g j(ζ j (s), )dµ(s) = i g j (t j, )dη j (t j ) by γ gj j. he assumption of independence implies that U i (g) = u i (a, σ i (s))g i (ζ i (s); da i )dµ(s)dπ j i γ gj j (a i). (11) i S i he definition of equivalent strategy profiles in the game Γ MW in Definition 1 is still valid here for the game Γ RR. Next, choose v i Meas( i, IR) such that v i (a, ζ i ( )) is the conditional expectation E(u i (a, σ i ( )) ζ i ). 13 hen, the property of conditional expectation implies that for any a i i, u i (a, σ i (s))g i (ζ i (s); da i )dµ(s) = v i (a, ζ i (s))g i (ζ i (s); da i )dµ(s) S i S i = v i (a, t i )g i (t i ; da i )dη i (t i ). (12) i i By equations (11) and (12), we obtain that U i (g) = v i (a, t i )g i (t i ; da i )dη i (t i )dπ j i γ gj j (a i) i = v i (a, t i )g 1 (t 1 ; da 1 ) g l (t l ; da l )dπ l j=1η j. (13) Π l j=1 j We shall now define a special case of the game Γ MW in Section 3 by taking 0 to be a singleton {t 01 }. he payoff function for player i is v i instead of the u i there in the definition of the game Γ MW in Section 3. he measure η on (, ) (Π l j=0 j, Π l j=0 j) is the product measure δ t01 Π l j=1 η j. he remaining elements of the model are the same as those in Section 3. For a mixed strategy profile g = {g i } l i=1, the expected payoff for player i is V i (g) = v i (a, t i )g 1 (t 1 ; da 1 ) g l (t l ; da l )dη, (14) which equals U i (g) by equation (13). hus, we obtain a special case of the game Γ MW that has the same expected payoffs as the game Γ RR. he following theorem follows obviously from heorem 1. heorem 2 In the game Γ RR, every mixed strategy profile has a strong purification. In comparison with Radner-Rosenthal (1982, heorem 1), the above theorem provides a strong purification result not only for an equilibrium but also for an arbitrary mixed strategy profile. On the other hand, a consequence of the DWW theorem that is available in DWW (1951a) (i.e., Parts (1) and (2) of Corollary 1) already implies that any mixed strategy equilibrium is strongly payoff equivalent to a pure strategy equilibrium; one can simply ignore the argument used to establish (iii) in the first paragraph of the proof of heorem It is easy to see such a function v i exists since is finite. 8

10 5 Large Non-nonymous Games Next, we turn to games with many players as in Schmeidler (1973). Let (,, λ) be an atomless probability space formalizing the space of players and = {a 1,..., a n } be the space of actions. payoff function u is a continuous function on M(). Let U be the space of payoff functions endowed with its Borel σ-algebra generated by the sup-norm topology. large non-anonymous game G is an element of Meas(, U). We shall also denote G(t) by u t, and since one can always rescale the payoffs, we assume without any loss of generality that there is M > 0 such that for all t, u t M. mixed strategy profile g (respectively a pure strategy profile g ) is an element of Meas(, M()) (Meas(, )). Let the distribution induced on by the mixed strategy profile g be denoted by γ g, where for all B, γ g (B) = g(t; B)dλ(t). For any mixed strategy profile g, the expected payoff of player t is given by n i=1 u t(a i, γ g )g(t; {a i }) = u t(a, γ g )g(t; da). he average payoff U(g) is u t(a, γ g )g(t; da)dλ. n equilibrium in mixed strategies g is an element of Meas(, M()) such that for λ-almost all t, player t maximizes her expected payoff. hus, for λ-almost all t, u t (a i, γ g )g(t; a i ) u t (a i, γ g )p i for all p {p IR n + : p i = 1}, i=1 i=1 which implies that if g(t; {a i }) > 0, then u t (a i, γ g ) u t (a, γ g ) for all a ; this means that u t (a i, γ g ) and u t(a, γ g )g(t; da) take the maximum value of u t (, γ g ) on. n equilibrium in pure strategies is simply a pure strategy profile g such that for λ-almost all t, player t maximizes her payoff u t (, γ g ). It is clear that for any pure strategy profile g, γ g is simply λg 1. n equilibrium in pure strategies g is a purification of an equilibrium g in mixed strategies, if γ g = γ g. he proof of the following theorem only uses a consequence of the DWW theorem that is already available in DWW (1951a), which is to say, Parts (1) and (2) of Corollary 1 above. heorem 3 ny mixed strategy equilibrium g for the game G has a purification. Proof: We also write u t (a, γ g ) as u(a, t, γ g ). pply Parts (1) and (2) of Corollary 1 to the collec- {(, ), λ,, u(,, γ g ), g} to obtain a pure strategy profile g Meas(, ) such that λg 1 = tion g(t; )dλ = γg and u t (a, γ g )g(t; da)dλ = u t (g (t), γ g )dλ. (15) For λ-almost all t, since u t(a, γ g )g(t; da) takes the maximum value of u t (, γ g ) on, we have u t(a, γ g )g(t; da) u t (g (t), γ g ). Equation (15) implies that u t(a, γ g )g(t; da) = u t (g (t), γ g ) for λ-almost all t. Hence, for λ-almost all t, u t (g (t), γ g ) = u t (g (t), λg 1 ) takes the maximum value of u t (, γ g ) on, which means that g is an equilibrium in pure strategies and hence also a purification of g. 6 Large nonymous Games Finally, we turn to anonymous games considered in Mas-Colell (1984) and Khan-Sun (1991). large anonymous game is a probability measure µ in M(U), where U is the space of payoff functions as specified in the previous section. he game is said to be dispersed if µ is atomless. i=1 9

11 n equilibrium τ of the game µ is an element of M( U) with marginal measures τ and τ U such that (i) τ U is µ, and (ii) τ(b τ ) = τ({(u, a) (U ) : u(a, τ ) u(x, τ ) for all x }) = 1. n equilibrium τ can be symmetrized if there exist h Meas(U, ) and another equilibrium τ s such that τ = τ s and τ s (Graph h ) = 1, where Graph h = {(u, h(u)) (U ) : u U}. In this case, τ s is said to be a symmetric equilibrium. heorem 4 Every equilibrium of a dispersed large anonymous game can be symmetrized. Proof: For an equilibrium τ of a dispersed large anonymous game µ, there exists g Meas(U, M()) such that τ(b) = U g(u; B u)dµ(u) where B is the Borel product σ-algebra on U and B u the section of B in. 14 Since τ(b τ ) = 1, it is easy to see that for µ-almost all u U, g(u; (B τ ) u ) = 1 and supp g(u) (B τ ) u. pply Corollary 1 to the collection {(U, B(U)), µ,, g} to obtain a pure strategy h Meas(U, ) such that µh 1 = U g(u; )dµ = τ and h(u) supp g(u) (B τ ) u for µ-almost all u U. Let τ s be the distribution µ(id U, h) 1. hen, τ s U = µ and τ s = µh 1 = τ, and B τ = B τ s. We can thus obtain that h(u) (B τ s) u for µ-almost all u U. Since τ s (Graph h ) = 1, we have τ s (B τ s) = 1, and hence τ s is a symmetric equilibrium that is a symmetrization of τ. 7 Concluding Remarks he primary motivation of this short essay has been to show that the general purification principle of DWW can be applied synthetically to games based on atomless measure spaces: one-shot, finite player games with incomplete information as well as one-shot large non-anonymous and anonymous games, all in a finite-action setting. In conclusion, we note that the theorems invoked and extended here, along with the strengthened purification concepts to which they have been applied, may have further application to more general settings; in particular, to pure-strategy equilibria in atomless Bayesian games with infinite action sets or over time; see Yannelis- Rustichini (1991), Balder (2002), Khan-Sun (2002) and their references. 14 For the existence of such a disintegration, see for example, sh (1972). 10

12 References sh RB (1972) Real nalysis and Probability. cademic Press, New York. Balder EJ (2002) unifying pair of Cournot-Nash equibrium existence results. J. Econ. h. 102: Dvoretsky, Wald, Wolfowitz J (1950) Elimination of randomization in certain problems of statistics and of the theory of games. Proc. Nat. cad. Sci. US 36: Dvoretsky, Wald, Wolfowitz J (1951a) Elimination of randomization in certain statistical decision procedures and zero-sum two-person games. nn. Math. Stat. 22:1-21. Dvoretsky, Wald, Wolfowitz J (1951b) Relations among certain ranges of vector measures. Pac. J. Math. 1: Khan M, Sun YN (1991) On symmetric Cournot-Nash equilibrium distributions in a finite-action, atomless game. In Khan M, Yannelis NC (eds.) Equilibrium theory in infinite dimensional spaces. Springer-Verlag, Berlin, pp Khan M, Sun YN (2002) Non-Cooperative games with many players. In umann RJ, Hart S (eds.) Handbook of Game heory. Vol. 3, Chapter 46. Elsevier Science, msterdam, pp Mas-Colell (1984) On a theorem of Schmeidler. J. Math. Econ. 13: Milgrom PR, Weber RJ (1985) Distributional strategies for games with incomplete information. Math. Oper. Res. 10: Radner R, Rosenthal RW (1982) Private information and pure-strategy equilibria. Math. Oper. Res. 7: Schmeidler D (1973) Equilibrium points of non-atomic games. J. Stat. Phys. 7: Wald, Wolfowitz J (1951). wo methods of randomization in statistics and the theory of games. nn. of Math. 53: Yannelis NC, Rustichini (1991) Equilibrium points of non-cooperative random and Bayesian games. In liprantis CD, Border KC, Luxemberg WJ (eds.) Positive operators, Riesz spaces, and economics. Springer-Verlag, Berlin, pp